IC/68/68

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

FINITE-COMPONENTFIELD REPRESENTATIONS OF THE

CONFORMAL GROUP

G. MACKAND

ABDUS SALAM

1968

MIRAMARE - TRIESTE

. • * * •*»••**» <..

IC/68/68

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FINITE-COMPONENT FIELD REPRESENTATIONS

OF THE CONFORMAL GROUP*

G. MACK **

International Centre for Theoretical Physics, Trieste, Italy.

and

ABDUS SALAM***

International Centre for Theoretical Physics, Trieste, Italy.

MIRA MARE-TRIESTE

July 1968

* To be submitted to "Reviews of Modern Physics".v * Permanent address> University of Munich, Fed. Rep. Germany.

*** On leave of absence from Imperial College, London, England.

ABSTRACT

We review work done on realization of broken symmetry under

the conformal group of space-time in the framework of finite-component

field theory. Topics discussed include: Most general transformation

law of fields over Minkowski space. Consistent formulation of an

orderly broken conformal symmetry in the framework of Lagrangian

field theory; algebra of currents and their divergences. Manifestly

conformally covariant fields and their couplings.

FIELD-KEPRESEHTATIOITS OP THE CONFOHMAL GROUP

I . INTRODUCTION"

Tie oonformal symmetry of space—time as a possible generalization

of Poincare symmetry has provided a recurrent theme in pa r t i o l e physics.

The problems associated with oonformal symmetry are i ) i t s physical

in te rpre ta t ion and i i ) the problems a r i s ing from i t s "broken character

and the precise manner of descent to Poincare invariance.

In this paper we -wish to concentrate on i i ) and review work

done on rea l iza t ion of conformal symmetry - and pa r t i cu la r ly of the

a lge t ra associate-..', with the group - using f i e ld operators which sa t i s fy

Lagrangian equations of motion. The f ie lds may "be defined over the

Minkowski space—time manifold x,*. or over a protect ive six-dimensional

manifold fl re la ted to x,< . We "believe t h i s approach to conformal

symmetry offers the "best hope of exploi t ing the symmetry physically in

contrast to approaches "based on a group theore t ic treatment of s t a t e -

vector spaces associated with the group. ' This i s e ssen t ia l ly "because

in such an approach i t i s extremely d i f f i cu l t to see how to "break the

symmetry down to Poinoare invariance.

The plan of the paper i s as follows: In Sec.II we give the most

general transformation law of f ie lds (defined over Minkowski space x ^ )

for conformal symmetry, using the theory of induced representa t ions ,

and also exhiMt the f i e ld rea l i sa t ions of the generators of conformal

alge"bra. In Sec . I l l we enumerate the Lagrangians (for pa r t i c l e s of

spin s ( l ) which are conformally invariant and describe some modes of

symmetry breaking - in pa r t i cu la r the physically i n t e re s t ing case of

conformal symmetry "breaking to the extent of "breaking d i l a t a t ion

invariance only, (An expl ic i t model i s discussed in the Appendix.)

Even where the formalism of Sees. I I and I I I i s conformally covariant

i t i s not manifestly so* In Sec.IT we t r e a t the manifestly covariant

formulation of wave equations for quantised f ie lds defined over a s ix -

dimensional protect ive space. Spins higher than one can "be more

easi ly t rea ted using t h i s formalism. In Sec.V we review an attempt to

understand V-A or V+A weak interact ion theory as a conformally invariant

theory of fundamental in te rac t ions . Hot considered in t h i s paper are

representa t ions of the conformal group which give r i s e to i n f i n i t e

component f i e l d s . They wi l l "be deal t with elsewhere.

- 1 -

II, TRAJTSFOHMATIQW LAW OP FIELDS

The conformal group of space time consists of coordinate transforma-

tions as follows:

1.Dilatations x1 = px , p >0

2.Special conformal transformations /__ n \n -j ^ 14.. x;

x- =o- 1( X)G V V2)

where o(x) = 1 - 2cx + c x

3» Inhomogeneous Lorentz transformations

Its generators D of dilatations, K of special conformal transfor-

mations, and P , M of the Poincare group admit of the following

commutation relations (C,R.):

[D,P ] = iP [D,M ] = 0

[D,K 1 = -iK [K ,K] =0 (II.2)

[K ,P ] = 2i(g D- M ) [K ,M 1 = i(g K - g K )•p. v °v-v liv *• p "jiv PU v pv u

plus ' hofie of Uie P o i n c a r e a l g e b r a . P a r i t y must sa t i s fy .

7T DTT1 = D , TT K TT* a *K , TT? TT*' = t i ; , T T ^H H V- U ( I I . 3 )

• here t.he + sign stands for M-=t), ^nd tht~ - .\;.ifTn for )i~lt,'!,jj.

The C.ii. (II.2) can be brought into .'1 form which exhibits ':he

0(2,4) structure of the conform*1 group explicitly by defining, for

J = M , J_, = D y J c = i ( P - K ) t J r = i ( P + K ) .V-v v-v ' 56 y 115 I V - V- no I H v-

f JKL' JMJ^ = ± (SKNJLM + «LMJKN ~ gKMJLW " SLNJKM } (II .4 )

where gAA = C+ ,-+) , A = 0,1,2 ,3,5,6

Note that the special conformal transformations do not take mo-

mentum eigenstates into momentum eigenstates, an [K ,P 1 does not

commute v;i'h the momenta P . Vie also notice the relation

iccD -ioD -a 2e r e = e r

Because of this relation, exact dilatation symmetry (vith an inte-

t;rable generator D that takes one-particle states into one-particle

states) iaiplies that the mass spectrum is either continuous _pr jn 11_

masses are zero. ' This clearly implies that exact

dilatation symmetry ie physically unacceptable and one will

-2-

•ch«y«£oa?« have to male* naaumptlons oa tb# dynamioa whioh

specify how the conformal symmetry is broken, A theory of this type,

which is in a sense analogous to the £U(3)xSU(3) current algebra

with PCAC will be presented in the next section.

First we have to define v however, what we mean by an(infinitesi-

mal) dilatation or special conformal transformation, as we want it

to transform a physical system into another one that is realizable

in nature (and not,e.g.,a proton wi Mi mass m into some nonexistent

particle with mass p m, for arbitrary p > 0 ) .

To do this we postulate that there exist interpolating fields to

every particle which transform according to a representation of the

conformal algebra, i.e.

for infinitesimal gsG(2,*f)

where g acts on the coordinates TC as indicated in eq.QQ.l)

It follows from eq.Q"J.5) and the multiplication lav; for the re-

presentation matrices T(g) that

S(g,O) must be a representation of the stability subgroup of

It is seen from eqjfll.l) that this subgroup (the little group in

physical usage) •• hir.h leaves x = 0 invariant is given by special

conform;il transformations, dilatations and homogeneous Lorentz

transformations. From the C.K. eq.(ll.2)one finds that thp Lie al-

gebra of this subgroup is isomorphic to a Poincare algebra + dilata-

tions, i.e. we have

The ^-dimension.: 1 translation subgroup T. corresponds to the spe-

cial conformal transformations, and S0(3,l) is the spin part of the

Lo r e n t z group«

Given any representation S(g,O) of the little groupfu.g) we

can nov.1 dete.t mine ,in accordance with the standard theory of indu-

ced representations^the complete action of the generators of the

conformal group on the field fix.) as follows:

Let Z , ' \. be t}ie infinitesimal generators of the little

group (II.6) corresponding to Lorentz transformations, dilatations,

-3-

and special conformal transformations^respectively. They satisfy

av

Qioose the basis in index space in such a way that

space time translations do not act on the indices, i . e .

fo l lows t h a t f o r every element X of the conformal n

X <p(x) = expC + i P x^) x'cp(O) where

XI = exp(-iP^cU) X expt+iP^)

~ j ^ j x . . . x [ P v , [ . . . Lpv , x J . . . J J

I nn«o

The important point is that the sum on the EHS. of eq.ClI. 9) i

actually finite. From the C.R. eq.(ll,2) it is found by inspection

that there are at most three non-vanishing terms in l.his sum.

Evaluating the finite multiple commutators,e.g.,for X a K^ , we get

From this we now deduce the action of K , D, M on <p(x), since the

action on fp(0) is known by hypothesis; e.g. K^/p(O) = tt (p(0). The

final results are

P q>(x) = -id (p(x)

, s r / , ^ •> r \ (11.10)

^v p. v v u ^v

D if>U) = {-i x av + S |«p(x)

K -p(x) = {-i(2x x dv - x2d + 2ixv[g S - Z

-4-

where the matrices I , &*%, satisfy the C.R, (2,7).

We have thus shown that all field theoreticallyadmissible represent-

at ions of the oonformal alge"bra are induced "by a representation of the

alge"bra of the little group (IT«_6)» Since this algebra has two non-

trivial ideals (» invariant subalgebra) [D] ijT.^T. there arise the

following types of representations*

I. finite-dimensional representations of the little group

a) V 0

II. infinite-dimensional representations of the little group.

Regarding these representations and their physical usea, the following

remarks are in order:

1) For case la) b = iX! by Schur's lemma if the Z!.,, form an irrreduc-

ible representation of the homogeneous Lorenta algebra.

2) In Sec.Ill it will be shown that the notion of a (broken) con-

formal symmetry admits of a perfeotly consistent formulation in the

framework of ordinary Lagrangian field theory. This theory makes use

of finite-dimensional representations of the little group (IT.6), with

•*,,=> 0 (type la). A H generators will be hermii:ian.

3) For case Ib) the conclusion that all the Tc^must be nilpobent

follows from the well-known fact that in any finite-dimensional re-

presentation of the PoincarS algebra the generators of translations

are nilpotent.

4) The possibility of using representations of type Ib) for physical

purposes is interesting "because it can give rise to spin multiplets.

The representations induced in this way are not fully reducible

however (and therefore not unitary representations -} cf. theorem 1 of

Sec.IV.2). Further discussion on this possibility will "be given in

section V, -where Hepner's work on the use of these representation

will be reviewed.

5) The possible use of infinite-dimensional representations of the

little group will not be discussed in this paper. This would lead

to the consideration of infinite-component field theories which will

be discussed elsewhere.

Ill, LAGRA3STGIAB FIELD THEORYj APPLICATION TO STRONG AED

ELECTROMAGNETIC INTERACTION

In the present section we shall show that the idea of an

orderly "broken symmetry under the conformal group of space time

admits of a perfectly consistent formulation in the framework of

ordinary Lagrangian field theory. The considerations presenteds)

here are an extension of an unpublished note by one of the present authors.

For simplicity we assume fields with spin ^1 and minimal couplings.

We shall show that:

1) There exist local conformal currents k and a

dilatation current ^)» such that the corresponding

generators K., and D are hermitian and hav« C.R.

with the particle fields as given in eq.(H.10) with

6 =» i f l , icjt = 0 (type la). This is true

independently of whether the action \ £d x is

invariant or not.

2) The kinetic energy term without mass is conformal

invariant. The same is true of all non-derivative

couplings with dimensionless coupling constants and all

couplings arising from (Yang-Mills type) gauge field

theories. This includes electromagnet ism. It also

inoludes weak interactions mediated by an intermediate boson,

if this boson is associated with a gauge field associated

with some internal group (e.g., for hadrons, the

Cabbibo SU(2) subgroup of one ofihe SU(3) ideals of

chiral SU(3) ® SU(3),or for the U(2) ® U ( 2 ) group

considered for leptona by Ward and Salam }

which includes both EM and weak interactions.!

3) Besides the exact symmetry limit corresponding to mass-

less particles only, the possibility also exists

of a spontaneous breakdown of conformal symnetry. There,

all particles can be massive except for 1 = 0 , T =0

massless Goldstone boson. An example of a corresponding

Lagrangian (the O" model of Gell-Mann and Levy ) is

discussed in the Appendix,

-6-

Abstracting from L&grangian field theory, a current algebra type

scheme may be set up. It is composed of the C.R, of the currents with

the particle fields, eqs.(III.l) and (III.6), and the relation(III,17)

between the divergences of the currents. In addition, an algebraic re-

lation between the divergence of the dilatation current and the axi'jl

vector currents of chiral SU(3)xSU(3) has been proposed elsewhere and

is given in eqs.(A.4) and (A,5) of the Appendix. Eq.ClII.17) expresses

the idea that the breaking of conformal symmetry is minimal in the sense

that there is only as much breaking of the conformal symmetry as is in-

duced by the breaking of dilatation symmetry alone.

1* The conformal currents

According to eq.(II.lO) we want to transform the interpolating

fields as follows

- I*. X

M t f

If <p is the electromagnetic vector potential, we may postulate this

transformation lav/ only up to a gauge transformation. Eq.(III.l) is19)to be understood in this sense in the following. The well known

reason J'or this is that, for a masslesn particle, the vector potential

is not a manifestly Lorentz covariant field in the ordinary sense.

-7-

Through the last equation, £,,v *s ^^fined in terms of the spin

of the particle *

When acting on a spin 0 field £ v = 0

when acting on a spin •§• field £«v - r *• > v^

when acting on a spin 1 field (Z v A) = * (<\ « -» - v A )

The present theory does not lead to multiplets of particles with

different spin, We fix the values of X to "be

H = _ \ for scalar and X.~ ~J_ for spin ~ fields

vector fields

(III.2)

This choice is necessary in order to obtain acceptable currentsthen

because only ithe canonical equal time commutation relations of "the

fields are invariant under dilatations. The values of I in (ill,2) agree

with te^dimension of length of the fields in question. Note that

(HI.lc) is of the form (f'(x) = cp (?" x' under x , -> ^x ; ?- i* e

so that the fields transform under dilatations according to their actual

dimension of length.

We can now write down the following local currents:

.3)

-8-

The angular momentum current has tbe usual form. The energy

momentum tensor is defined by

V,

It is convenient to choose the kinetic energy term in the Lagrangian

ass

- ( "i 3 - E ")l + 1 m* 3V BV

(III.4)

Sinoethe zeroth components of the currents, T0JU , 3) , kOju and

are all hermitian , the corresponding generators, formally

defined as the space integrals of the zeroth component of these

currents,are also hermitian.

One checlcs t>y using the canonical equal time

C.R. of the fields

that

»»% r ° (III.6)

satisfy eqs,(lll.l). This is independent of whether the Lagrangianis

conformal invariant or not, (The integration in (III.4) goe3 over

some volume v including z = x•,)

-9-

2. Divergence of the currents

We now turn to the discussion of the properties of the

divergence of the dilatation current 3) y and special conformal

currents K^.t . The dynamical information of a symmetry (exact

or "broken) defined by the transformation law of the fields, lies

in the properties of the divergence of the corresponding currents.

From eq, (ill,3) it is seen that the dilatation current

*3J V(I) depends on x explicitly and can therefore not "be coupled toa,

the field of^vector particle. However, using energy momentum

conservation d ' Vp = 0» w e find for its divergence

V2> (x( - Tvv - \ { £ ^ - ~ ~ - * T }

From this we see that:

The divergence of the dilatation current is a local

1 = 0 , J~T ^ 0+ field .

U3ing the Euler Lagranp© equation of motion , (ill,7 )

can be rewritten in the form

From eq.(ill.8) it is seen that all pieces of the Lagrangian thatnonzero

do not involve constants with t dimension of length give zero

contribution. To see this, notice that the EHS of eq.(lll,8)

vanishes ) since it is simply the Euler equation for a homogenous

function (all terms having the same dimensionality). This establishes

that the kinetic energy Jterm without mass is dilatation invariant^

and so are all couplings with a dimensionless coupling constant.

Fext we turn to the divergence of the conformal currents.

Using energy momentum conservation we find from eq,(111,3)

-10-

P vThe last line is equal to 2x d Trf and vanishes by angular momentum

conservation* Hence

V h " 2V ^ ^ + V (III. 10)

where V is a local vector field defined by the second line on the RHS

of eq.(III.9). From eq.(lll.4) one checks by explicit calculation that

the kinetic energy term without mass gives zero contribution to V ,

Since it gives no contribution to J S) either, we see that the kinetic

energy term without mass is fully conformal invariant. Furthermore, from

(111*10) the condition that an interaction Lagrangean Jfa be fully con*

formal invariant is found to be the following

l) It is dilatation invariant, i.e., has a dimensionless

coupling constant

Condition 2) is independent from condition 1) as is seen from the ex-

ample Zx - gA^it(d a) which satisfies l) but not 2). An example vhich

satisfies neither l) nor 2) is the derivative pion nucleon coupling

NJT y Nd it . For nonderivative couplings, condition 2) is trivially

satisfied.

3. Yang-Mills theory

Let us now turn to the question of the conformal invariance

of the coupling of vector gauge fields in a Yang-Mills type gauge fieldis) • a.

theory. Let A be the internal n-parameter symmetry ilgebra, and B »

a = 1...n , the corresponding gauge vector fields. Under an infini-

tesimal transformation with constant infinitesimal parameter e*, all

fields transform according: to

-11-

(III.If?)

where the matricesT form a hermitian representation of the algebra

A, Henaiticity reads

t » \* -r A

^V* ; * a<* (III.13)

As is well known ' , all couplings of the vector fields B* are

completely determined from the postulates of a Yang-Mills type theory

and are obtained by the substitution

\- *K ~* \ A " l3 Ta A ^ ^ (in.14)

Here, g is t dimensionless real coupling constant. To test for con-

formal invarianoe we see that condition l) above is always fulfilled,

while condition 2) is also satisfied beoause the only derivative

couplings are the couplings of mesons, which have the form

(ITI.15)

for spin 0 fields C> , and

for spin 1 fields <f ^ » Inserting into eq. (HI . l l ) and making use

of eqs,(111.13) and (il l .2) one finds that condition 2) is indeed

satisfied.

It is now tempting to speculate that in physics there is only as

much breaking of conforms! symmetry as is induced by the breaking of

dilatation symmetry. In other words there should be a remainder of

conforraal symmetry in the sense that all couplings satisfy condition

2) above. This is equivalent to the algebraic condition

¥

-12-

The virtue of this restriction is that It still allows for a "breakingof th« symmetry by the maaa terms in the Lagrangian.

IV. MANIFESTLY 0(2,4) C0VAHIA8T FIELD TRANSFOEMATION LAW

General experience from the history of elementary particle

physics may lead one to the opinion the "the only good covariance is a

manifest covarianoe". This is the motivation for the present section.

It is mainly pedagogical in character and much of the material

presented msy "be found in the literature for special cases, "but is

presented here in a unified way. Manifestly oonformal Invariant freelit)

field equations were first discussed by Dirac, Invariant

interactiona were given by Kastmp.

The problems to "be solved are the following:

1) Write down manifestly conformal invariant

transformation laws for fields,

2) Determine the relation between the old fields <£/ (* /

ocourring in the transformation law eq.(H.lO) and

the new fields which are transformed manifestly co-

variant ly,

3) Write down manifestly invariant free field equations

and interactions.

The new fields will be multispinor functions on the four-

dimensional surface in a five-dimensional protective apace rather

than Minkowski space. Their physical interpretation will nevertheless

be guaranteed by correspondence with ordinary fields y> (x) over

Minkowski space disoussed in Sec.II, This correspondence also allows

one to oonsider questions of unitarity and quantization "by reference

to Minkowski spaoe*

-13-

1, Manifestly covariant transformation law for fields

A manifestly 0(2,4) covariant transformation law may be

written down for multispinor functions "X (») defined on the five-

dimensional hypersurfaces ofR given "by

^ " L" (17.1)

and satisfying "Xp*1?1 " ^/i'"**1 •

Summation over B is over^l,2,3,5,6 with metric (+ ; -+). There

are three essentially different surfaces, corresponding to

L2 - ± 1,0.

Suppose that •J Y*-D ^S any representation of the algebra

of 0(2,4) ( fcSU(2(2)) acting on the indices of % * (h) only. Then a

manifestly covariant transformation law including an orbital part is

given by (of. eq.(II.4))

(17.2)

where

Clearly L,n and •§Y,-n commute with each other and satisfy the C.R. of

0(2,4) separately* It is important to notice that L „ is a well

defined operator when acting on functions that are only defined on

the hypersurface (IV.l). This ia so because L ^ corresponds to an

infinitesimal coordinate transformation which is a pseudorotation of

the hyperaurface (V.l) into itself.

The coie ^ ?* . 0 (17.3)

is also left invariant by the coordinate transformation ^ g ^ ^ ^ c > y ®

Moreover,this transformation commutes with the 0(2,4) rotations.

Therefore we may require the fields to be homogeneous functions on

the cone (IV.3)

i.e. {\(IV.4)

-14-

Thess homogeneous functions then depend arbitrarily only on

4 of the 5 coordinates which determine a point on the cone (V,3)»

i.e., just as many as there are coordinates in Minkowski space. We

shall restrict our attention to this oase in the following.

-15-

2* Mathematical preliminaries

Before proceeding we need to know a few mathematical

lemmas«

Lemma 1; A Bet of commuting, nilpotent(finite-

dimensional matrices X,^ can simultaneously be "brought to tri-

angular form with zeros on the diagonal by a suitable choice

of basis, '.ev

for all f& simultaneously

(IV.5)

This is a corollary of Engel's theorem which may be found in

standard textbooks. Recall that nilpotency of a matrix x « means

that there exists a positive integer m such that

Suppose that we are given a representation of the form

(II.10) induced by a finite-dimensional representation 1L , /<v, &

of the algebra of the little group (II.6). As we have seen in

(II.10)f, there arise in this way two types of induced representations.

x u o 0 (type la), and */*.4 ° but nilpotent (type Ib). By virtue of

lemma 1 we may assume in the latter oase that the four matrices X^i

are all of the triangular form (IV,5) without loss of generality.

Lemr-a 2i Induced representations of type Ib have an

invariant non-enipty sub space a\ on which an induced

representation of type la iB realized. This invariant subspace is

spanned by those components of the field Jf'(x) which satisfy

There is,however, no invariant complement to

invariant subspace.

The fact that the subspace defined by (IV.7) is non-empty

follows from(lV.5) because the top row of all the matrices K a is

-16-

identically zero. The subspace (IV,7) is invariant by virtue of eq.(ll.io)

and the C.R, (11*7). Finally, let 9 (x) be such that, for some •ixed 11,

TC <p (x) j£ 0 but TC ic <p (x) - 0 for all v. Such a <p exists by virtue

of (IV.5), Clearly <p ^ X*. Consider now K^ = exp(-iP x u) Ky exp(iP x^).

This is an element of the conformal algebra for arbitrary x (cf. Sec.II).

We have K ip (x

complement of 31.

We have K ip (x) = * » (x) e "H . Hence there cannot exist an invariant

Theorem 1* The induced representations of type Ib as des-

ribed in section II are not fully reducible.

At a heuristic level this is a corollary of the last statement

in lemma 2 s

We &lsc need some properties of the finite dimensional represen-

tations of the algebra of 0(2.4).

> Theorem 2t All finite dimensional representations of the

algebra of 0(2,4) without parity can be obtained by reducing out ten-

sor products of the two inequivalent four-dimensional representations

A and A given by matrices i as f ollov;s:no

V"

All matrices satisfy \ ^ ^ 0 • ^ (IV.8)

C+) (-1 )

Parity transforms A into A and vice versa . H^ are Diracmatrices and )S = Y !f y, y, ,

For a proof of this theorem see, e.g., ref. I . The theorem

essentially states that all finite dimensional representations can be

constructed out of fundamental representations. In the above, two fun-

damental representations are used, one corresponding to righh-handed

spinors and the other to left handed spinors.

-17-

The simplest non-trivial representation of 0(2,4) with

parity is eight dimensional and unique up to a choice of basis. It

is given by A •» ( A(~^ )• •"•* is.,however, convenient to make a

basis transformation so that

Y . * r Y v 1 • v

Parity is represented by «Q here. ^ are Pauli matrices. In

this form the eight-dimensional representation has been given by

thMurai, For this representation a Clifford algebra fl. exists

such that

1 t . p J ' 29*» (IV.IO)

These matrices (?* transform as a 6-vector under - Vji-g* This is

important for constructing invariant couplings. Explicitly, the fr.

may be given by

There exists also a conformal pseudoscalar

All matrices (V.9) satisfy

Finally we need the following corollary of theorem 2.

Corollary: Let 2,^ be any finite-dimensional irreducible

representation of the algebra of SL(2,C) extended to a representation

of the algebra (II,7) *>y choosing S« <( 1 , x. « o .

-18-

a4

Then there exists a finite-dimensional representation •§• Y,-Q °f

the algebra of 0(2,4) with the following property! There exists a

subspace 7t of the representation space on which

= v*?Vfor a l l e £ ?V and some su i tab le rea l m.

This coro l la ry guarantees tha t every f ini te-dimensional

representa t ion of 'the algebra ( I I . 7 ) with K, , = 0 can be extended to

a representa t ion of the 0(£,*!•} algebra ^ enlarging the representat ion

space. Not© tha t a t rue enlargement i s always necessary, unless

Z ^ v = 0, since no generator IC^ of a simple Lie algebra can be

represented by 0 in a n o n - t r i v i a l r epresen ta t ion .

Our representat ion •§• Y._ (whose existence i s guaranteed by

the corol lary)can be constructed sa follows: As i s well known, a l l

f ini te-dimensional representa t ions of SL(2,C) can be constructed from

left-handed and right-handed sp inors . Let the representa t ion spaoe

of Zj^^be constructed in terras of Lorentz two-component sp inors .

Then one obtains the desired representat ion space for 4Y simply by^ - component ^ AB ''

substituting oonformally transformingtspinors for the Lorentz spinora.The desired subspaoe is as defined by (IV,13c).

-19-

3. Relation between manifestly oovariant fields and fields

over Minkowski space.

Suppose we are given a field with indices, X(i), over the cone

(IV.3) which satisfies the homogeneity condition (IV.40 end transforms

according to eq.(IV.2), with jf^-a being a finit&dimensional

representation of the 0(2,4) algebra.

We want to obtain from this

a field tp(x) over Minkowski space which transforms according to eq.

(II.10). As we have seen, X(<j.) depends arbitrarily on 4 coordinates,

i.e. as many as there are space-time coordinates x . We will proceed

in three steps:

1. coordinateiransforraation w-»-x.

2. x-dependent basis -transformation in index space to transform

away the intrinsic part of the translation operator, i.e.

ensure eq.(Il,8).

3» Project out unphysic;?l components if necessary.

By step3 we mean the following:

After having carried out steps 1 'md 2 we shall already have arriv-

ed at a field over Minkowski space which transforms according to

eq. (11.10). If we start with a finite-dimensional representation

I*AB ° t h i s w i l 1 b e a representation with M = 1(Y ^-T ) / 0. If

v,re vant a representation with % = 0 (i.e. type la) we must project

onto the invariant subspace on which this is true, i.e. ve keep as

physical components only those which satisfy

*^ ¥ (x) s 0 for }i=0...3 , with •* s 1 (V g- r ) (IV. 1

This subspace is nonempty and invariant by lemma 2 of section IV.2.

Bq.QV. 14) may also be read as a subsidiary condition which makes the

unphysic^l components equal to zero. It may be necessary to emphasize

the conformal invariance of eq.QV.l4), It does not break down thenecessary

symmetry but is a/condition for the irreducibility of the represent-

ation .

Step 1 has been described in great detail by Dirac . For the

spin j case, step 2 has also been carried out by Dirac, and later

desoribed in greater de-bail by Hepner.n) we will give a unified

treatment for general spin.

-20-

Step 1. Define

a v 6 a % r^r15 16

The function (pa defined in this way does indeed only depend on x

and not on •jp+'^by virtue of the homogeneity condition (lV. f).

?g- tc is not an independent parameter anyway "because of eq.,Qv.3).

A conformal transformation GV.2) of X induces on fp a transforma-

tion of the form

where * • / a \Lliv " l ( V v * V u }

(IV .16)x 0 - x 6

p. v >

L r~ Llib

In particular we have then

PH 5Cx) = C-i^+T^) 5(x) where y^ ^ V e * ^ *

Step 2 . Define i l > J l )

<Po(x) = VaB ^ f l (x) where V = expCixUy^') (IV.17)

The mai.rix V exists bee-: use v;e assumed the V, finite dim^n-

Moreover, V is alwayst+>a finite polynomi-1 in x because Lhe Vt+>are also nilpotent. It

could therefore be worked out explicitly in each case, in practice

such straightforvari but sometimes tedious calculations can usually

be avoided by using translation invariance and the fact that V = 1

at x = 0,

Because all V commute, V has an inverse given by

Furthermore

v(-ic jv"1 = -ie -v<+) . ,^ v- u (iv.19)

Using eqs.GV.19) and the C.R, of 'he matrices ^ t. „ r,s given by

eq.CII.'O one may check, that the components of i.he field <p in the

nev; basis do indeed transform according to eq. (11.10) with

-21-

n is given by eq.QV.A). The remaining matrices have disappeared

from the transformation law.

Summing up, the sought-for relation between the fields <p(x)

andX(?) is given by eqs.flV.15) and (IV.l?). This establishes our

claim that* after having carried out steps 1 and 2, we arrive at

a field which transforms according to eq.Gl.10), with -K ^ 0 unless

As explained above we may thent as our step 3, proceed to dropp-

ing the unphysical components which do not satisfy eq.GV.l^)

The "dimension of length" ariaing in eq.frH.l) is related to the

degree of homogeneity n by

4 = n -i»(eigenvalue of {1'g in the subspace OV.!^)). (IV.21)

-22-

**•. Invariant wave aquations and interactions.

With manifestly covariant fields it is straightforward to vrite

down manifestly invariant wave equations and interactions. The follow-

ing examples are due to Kastrup. tf".B is finite dimensional

p.nd all fields are to correspond to fields over Minkovski space which

transform according

ed in section III.

transform according to eq.0l.10) with it = 0 , as do the fields employ-

The spin 0 field (scalar or pseudoscalar) corresponds to a con-

formal scalar A(>j) with degree of homogeneity n=-l, i.e.

^ Be BA( 7)= -A(%), *ABA(*> = 0. , (IV .22)

The free wave equation is :

OgA(?) = 0 where Qg = 6 dB (IV.23')

As discussed by Dirac , ' this is a veil defined equation for A(^)2 and only if

defined on the cone ^ = 0 only, if j[ n =-1 as we assume. By eqOV, 12),

t = n s-1 in agreement with the discussion in sectionlll.

The scalar field in Minkowsi space is then given by

a(x) = (?5+76)+1A('z) (IV .24)

and satisfies ^^&{x) = 0 . (IV.25)

The spin j field is an 8-coraponent spinorXC^) of degree of

homogeneity n = -2

B , rAfi given by eq.(lV.9) (IV.26)

The adjoint is defined by

* = X ro Tl (IV.27)

cf. eq.(IV,,12)

The corresponding 8-spinor over Minkov.ski space is again given

by eqs.GV.15) and (1V.17) •. hich reads

K ) C ) * 2 ( ^ + w h e r e x+= JC^+iTg) (IV.28)

Its physical components are given by eq.GV.l4) which takes the

simple form

(1+ T 3 ) V(x) = 0 (IV.29)

In the basis where T, has the usual diagonal form, these are just

-23-

the lowast four components. From eqs.dV.9) ftnd ffV.ai) one

finds ^ 3 n + j = -{ , as was assumed in section til because of

unitarity requirements.

The free wave equation is

A B = 0 LA» * ( 7 * ^ " ? » \ ) (IV.30)

This amounts to diagonalizing the second-order Casimir operator

i" JAB (Hepner and Murai).

The spin 1 gauge fields are 6-vectors Ag(^) of degree of homo-

geneity n= -1 .

*\ V * > = - AC (* ) ? <i*AB A )C = i(*AC AB " SBC AA ) t17'31)satisfying the subsidiary oondition 7* A £ ( ) « o . (IV»32)If we impose in addition the generalized Lorenta condition

u = 0

then the admissible gauge transformations are, for the electromag-

netic potential,

(IV.33)

where the gauge function S must be specified on a whole neighbour-

hood of the cone ^ = 0 , and satisfy there

2BdB S(^) = 0 ; Qg SC7) = 0 .

The free field equation is then

a6 Ac(t) = 0 (iv

Again, the choice of n=-l makes this into a well-defined equation

for A^Cij) defined on the cone •£ =0 only.

The corresponding field a-g(x) is again, given by eqs.flY. 15) and

(l»917/r For the first four components this takes the explicit form

and the subsidiary condition eq.QV. 32) reads

a6(x) - a5(x) = 0 (IV.36)

This can "be seen in the following way: For x = 0, we have

«j = 0 and *is » 16 . Therefore eq.CIV.iO is clearly true st this point.

Now eq.dV,ll) is conformal invariant and therefore,in particular,

translation invariant. Hovever, by construction all a^(x) transform

-24-

under translations according to eq. GJ.IO). Thus ., by translation in-

variance the validity of eq#(IV»36) for arbitrary x follows from its

validity at x =0.

The first four components of a~(x) are the physical ones as they

satisfy eq, (17,1*0 by virtue of the subsidiary condition eq.GV. 3*).

Namely

i ^ - V ' v 5 i gHv(a6"a5) = ° v.0,.,3 (IV.37)

Invariant, couplings»o on formal invariant

Following KaBtrup, it is easy to see that a /coupling between

a pseudoscalar field A(^) and the spin j field X(|j) is given by the

following wave equation*.

AH f-i(T LAB+if) X = g-^flcfl7XA

and the coupling of the electromagnetic field to the spin ^ field is

given by

where

= q jc(t) (iv.39)

The IJ-matrices are given by eqs.(IV,10) and OV.ll).

As is seen, the electromagnetic coupling is obtained by making

the gauge-invariant substitution 3_-»d-, - iq A . In this form it can

be immediately generalized to arbitrary sets of gauge fields Ac

Let T "be the representation raatricea of the relevant group as dis-

cussed in sectionllljthen the general rule is to substitute

in the free field equations. Summation over a is understood. In this way

one obtains couplings which are both conformal invariant and gauge

invariant.

Finally there also exists a cjuadrilinear conform." 1 invariant spin 0

boson coupling, A corresponding vave equation vould be,e.g.,

cr6 AC?) = g [ A ( ^ ) ] 3

Of course all couplings mentioned above can oocur simultaneously.

-25-

A point w« want to stress is the following: Not all manifestly

covariant looking couplings are physically acceptable. They must se*-

isfy the following additional requirements:

IV The field equations must have solutions which are homogeneous

functions. This requires that all terms in a certain field equation

must have the same degree of homogeneity* The degree of homogeneity

of such a term is calculated by counting each explicit coordinate *?

with +1, each derivative d/d*J, with -1, and each field with the ap-

propriate number n (e.g. -1 for bosons and -?. for fermions in the

cases discussed above).

2). The interaction terms must not couple unphysical field components

to physioal onea« This turns out to be a strong restriction in practice.

The couplings given, above do satisfy this condition, while, e.g.^ a

coupling XX A would not.

There is on easy way to check vThether condition 2 is satisfied

without going through the tedious transformption.s of section IV.J>.

Because of translation invarisncc, it is sufficient to check that

i-he condition is satisfied at x = 0 . This corresponds r.o = 0

ond \ =t&* At this point the boost operator V in eq.(lV,17) is simp-

ly unity: V _ = $ . Therefore we have in general

(IT .42)

where the abbreviation X(Q) = X( y = 0»7*m?<) has been used. The

physical components of a field at this point are then simply those

satisfying

i.e. for a spin i field Cl+x )X(0) = 0 (IT.44)

and for the electromagnetic field the four components A (0),

-It is now easy to check whether there in a coupling of phyrsic^l

to unphysical components or not. For example, in the cose of the

pseudoscalar coupling QV. 38) we find from eqs. QV, 9) • . • (IV.12 )

Thus we see that the pseudoscalar field A is only coupled to the

physical components of X(Q) vhich satisfy eq. (IV. *tfr). The one and

-26-

only component of A(0) la clearly physicdf It satisfies eq, 0

trivially. One can also check in this way, from eqs.(I7.42)

and CRT. W , that the wave equations (JV.38 ) and QV.39) do indeed cor-

respond to the Dirac •, Klein-Gordon and Maxwell equations for the phys-

ical field components in Minkowski space, with minimal electromagnetic

interaction and nonderivat'ive pseudoscalar pion-nucleon inter-

action.

In section HI an alternative characterization of all (physical-

ly acceptable) conform?! invariant couplings for spins 1 has been

given. If all vector mesons are assumed to be gauge fields, then

the manifestly conformal invariant couplings given above, and their

obvious generalization to the case of several fields of the same

spin, exhaust all poBnibilities for spin gl. This may be checked

by enumerating all possibilities, ae there are only a

few types of couplings with dimensionless coupling constant, and

the gauge field couplings are fixed in their form.

-27-

V. REPRESENTATIONS INDUCED BY FINITE-DIMENSIONAL REPRESENTATIONS OF

THE LITTLE GROUP (II.7) WITH K,, I 0 .r

As haB "been mentioned at the end of Sec.II, these representations

are interesting in principle because theycou&L give rise to spin multi-

plets, but they are not fully reducible (and therefore not unitary re-

presentations) The only author who has recognized the power of re-

presentations of this type is Hepner who uses conformal invarianoe to

generate uniquely Hhe V-A (or V+A) weak interaction. Assume we are

working with a four-component spinor V (quark, ju or e-field). If we

postulate that a four-Fermi interaction be conformal invariant, then

there exist two possible interactions

g (fi Y ^ V

corresponding to XXXX and t|AX)( % t| X Y QB in the six~din<en8ion-al language of Sec.IV.

Here £ V^.y either -\ d^Ys) or i^-*^)

depending on which of the two inequivalent if-:)im';nsional repre-

sentations Aft) of the algebra of the index-0(2,4) is chosen (cf.

Sec, IV eq.(lV.8)). We will set * = 1 numerically. This can al-

ways be achieved bya basis transformation with the matrix exp(iaV^),

for suitable a.The expression (V.l) is just the familiar V-A or V+A

coupling.

An invariant wave equation would be

The quantity n appearing here and in eq.. (IS. 20) must he a solution

of n - n + 2 m 0 in order that (V.2) be invariant. (This is related

to the homogeneity requirement discussed at the end of Sec.IV.)

-28-

We now break the oonformal symmetry and desoend to Foincare in-

variance by adding the symmetry-breaking term « JL'*1 **" - * (j -1) Y . j

eq.(V,2) goes over into the usual Dirac equation with weak interactions.

The point of view regarding weak interactions taken here is

different from that of Sec.III. The theory of Sec.Ill has the

advantage that the kinetic energy term without mass does not break the

eonformal invariance so that the canonical equal time C.R. of the fields

are conformal invariant and one can write down hermitian generators.

-29-

APPKTTOIXTHE or-MODEL AS AN ILLUSTRATION OP IDEAS IN SEC.Ill

Consider the Lagrangian of the tf-model of Gell-Mann and Levy n^

£ = N(i^-M)N + igN^TKj + j(c^ite>VuV)

(A 1• i(^oartf - [/+ ^i}^ ) -xt^S* 1} - i o ^ + a

%\ ) .

Here N is the nucleon field, n is the pion field, and a is the field

of a 1=0, J =0 meson, f= g/2M. Let us choose the free parameter X to

4Re«xpressing !.he Lagrangian in terms of the field a% =

and calculating the dilatation current and conformal currents

from eq,(nX>3), one finds for their divergences

* V g - W a(x) (a), 6 v^ = 2 %oVa v (b) (A>2)

The last equation follows directly from eq.OHL.10)ff since the pre-

sent Lagrangian does not involve any derivative couplings, m is

the (bare) or-mass. We see that in the limit of a massless boson n

both currents are conserved, and we have a spontaneous breakdown

of conformal symmetry.

With the usual definition of the axial vector current 01. forJ

this model, one finds that generally, also for m ^ 0

= -iS..f ( 6 ^ - j / f " 2 ) for i,j = l,2,3

(A-3)

Elsewhere it haa "been proposed to generalize this formula to

ohiral SU(3) x SU(3) in the following form: 7)

^ 3 ^ = aouo(x) + a8 u8 ( x ) "

with a + i a 8 = . 3 ( F

^$/MQ is a measure of the breaking of the eightfold way. The u.

must satisfy the C,R. of (integrated) scalar densities with vector33)

and axial vector currents as proposed by Gell-Mann (i,j,k = 0...8)

-30-

where F ^ ^ d 3 x

= " i V k vk ( x ) (A'5) '

with

v.(x) = 9U0L Hx) for j=l ,2 ,3 .

The matrix elements of u + jru8 a r e ^ o w n in. current algebra calculations

as cr-terms, A method to caloulate them on the "basis of eq.(A*4) and eq.

(III.1) has been outlined in Ref.7.

- 3 1 -

FOOTNOTES AND REFERENCES

1) A historical survey including an extensive list of literature prior

to 1962 is found in Refs. 2 and 3 below and in the review article

by T. Fulton, R, Rohrlich and L. Witten, Rev. Mod. Phys. M,

442 (1962).

2) H.A. Kastrup, Ann. Physik^, 388 (1962).

3) F. Gursey, Nuovo Cimento 3, 988 (1956).

4) Discussions on the physical interpretation of the conformal group

of space-time are found in:

A. Gamba and G. Luzatto, Nuovo Cimento 18, 108 6(1960);

E.C. Zeeman, J. Math. Phys. 5, 490(1964);

T. Fulton, R. Rohrlich and L. Witten, Nuovo Cimento .26, 652

(1962) and Ref. 1;

L. Castell, Nucl. Phys. B4, 343 (1967); B5, 601 (1968); Nuovo

Cimento 4J5A, 1 (1966).

J. Rosen, Brown University preprints NYO-2262 TA-151,

NYO-2262 TA-161 (1967);

H.A. Kastrup, Phys. Rev. 142, 1060 (1966); 143, 1041 (1066);

150, 1189 (1966); Nucl. Phys. 58, 561 (1964) and Ref. 2,

plus a list of publications prior to 1962 reviewed in Ref. 1 and 2.

Without further discussion of the usefulness and consistency of

possible different interpretations, we adopt for the purpose of

the present paper the following point of view, following essentially

Kastrup:

1. Exact dilatation symmetry would imply that to every

physical system in any given state sub specie aeternitatis

and for arbitrary p > 0 another one exists which is realizable

in nature and differs from the first one only in that every physical

observable is changed by a factor p ; where X is the dimension

of length of the observable in question.

2. Special conformal transformations may be interpreted as

space-time dependent dilatations.

- 3 2 -

3. We consider only infinitesimal transformations. Then

no problem ar ises with causality, in particular the equal t ime

C. R. of fields will be invariant. (Sec. III. )

5) Unitary irreducible representations of SU(2, 2) (^ O(2, 4))have

been constructed by:

Y. Murai, Progr. Theoret. Fhys. (Kyoto) j), 147(1953);

A. Esteve and P. G. Sona, Nuovo Cimento 22_, 473(1964);

I. M. Gel'fand and M. I. Graev, Izv. Akad. Nauk SSSR, Ser. Mat.

29, 1329 (1985) and "Proceedings of the international spring

school for theoretical physics", Yalta (1966);

A. Kihlberg, V. F. Miiller and F. Halbwachs, Commun. Math.

Phys. Z, 194 (1966);

I. T. Todorov, ICTP, Trieste, preprint IC/66/71;

R. Raczka, N. Li mid and J. Niederle, J.Math. Phys. 1_, 1861,

2026 (1966); 8, 1079 (1967);

T. Yao, J. Math. Phys. 8, 1931 (1967).

The problem of reduction with respect to the Poincar6 subalgebra

has been considered for a few special cases in:

R. L. Ingraham, Nuovo Cimento JjJ, 825(1954);

L. Gross, J. Math. Phys. j>, 687 (1964);

L. Castell (1967) loc. cit.4> ;

B. Kursunoglu, J. Math. Phys. 8, 1694 (1967).

6) J. E. Wess, Nuovo Cimento 1JJ, 1086(1960).

7) G. Mack, Nucl. Phys. B5, 499 (1968).

8) G. Mack, Ph.D. thesis, Berne, Switzerland (1967).

9) G.W. Mackey, Bull. Am. Math. Soc. §£, 628 (1963);

R. Hermann, "Lie group for physicists", ed. W.A. Benjamin,

New York (1966),ch.9.

10) The finiteness of this sum follows also from a general theorem

of O'Raifeartaigh: Let G be any finite-dimensional Lie algebra

containing the Poincare' algebra P=SO{3,1) ;g> T . Then

-33-

there exists a finite number n_ such that

[T, [T, [. . . [T, X]. . . ] ] ] = 0 for n > n and any X £ G .i _ ^ i U

n times

In the present case n = 3 .

L. O'Raifeartaigh, Phys. Rev. Letters .14, 575 (1965).

11) N. Jacobson, "Lie algebras", Interscience Publishers, New

York (1962), p. 45, corollary 2.

Recall that nilpotency of a matrix H means that there exists a

positive integer m such that M. = 0 .

12) W.A. Hepner, Nuovo Cimento .26, 352(1962).

13) Minimal couplings are those involving no derivatives of fermion

and up to first-order derivatives of boson fields; they have the

virtue of not affecting the canonical equal time C. R. of the

fields.

14) Thi- conformal invariance of the free Maxwell equations and the

free massless Klein-Gordon, Dirac and Weyl equation has been

known for a long time, cf. Refs. 1 to 3.

A conformal invariant spinor equation with a non-linear interaction— 2/3 1)YY') ' has been considered by Giirsey .

Free wave equations in Minkowski space for massless particles

with arbitrary spin have been considered by:

J.A. McLennan, Nuovo Ciraento 3, 1360(1956); _5, 640(1957);

L. Gross, loc. cit.

Gross also presents a proof that the solutions of these equations -

and in particular the Maxwell equations - span a Hilbert space

on which a unitary representation of the conformal algebra of

space-time is realized.

15) C.N. Yang and R. L. Mills, Phys. Rev. 9j>, 191 (1954).

16) N. Cabibbo, Phys. Rev. Letters H), 531 (1963).

17) Abdus Salam and J. C. Ward, Phys. Letters L3, 168(1964).

18) M. Gell-Mann and M. Le"vy, Nuovo Cimento 1J3, 705(1960).

-34-

19) S. Weinberg, Phys. Rev. 134, B882 (1964); 138, B988 (1965);6)

see also J. E. Wess, loc. cit.

20) E.g. , for baryons this is seen as follows: If the generators

are hermitian, then the transformation law for y implies that

for"yf. Then

- ? 6 U - JJ •

This is of the form (III. 5) only if X - - ~ . If a different value

for X is chosen, there exist no hermitian generators that induce

(III. 1). The value £ ~ - — also follows from the requirement

that the free massless Dirac equation be conformal invariant.

Similarly the free massless Klein-Gordon equation for a spin-

less field is only conformal invariant if X = - 1 .

21) For the free field case with exact symmetry, similar currents

have been written down by :

it.1

6)

14)J. McLennan, loc. cit. ; and

J. E. Wess, loc. cit.

22) D.M. Greenberger, Ann. Phys. (N. Y. ) _25, 290(1963).

23) R. Utiyama, Phys. Rev. l_01, 1597(1956). Our matrices Ta

differ from his by a factor of i .24) P.A.M. Dirac, Ann. Math. J37, 429 (1936).

25) H.A. Kastrup, Phys. Rev. 150, 1186 (1966).

26) A subtle point here is that the points where at least one co-

ordinate Xu is infinite may also correspond to finite co-

ordinates on the cone, see for example Ref. 27. This is, how-

ever, not important here as we consider only infinitesimal

transformations.

4)27) L. Castell (1967) loc. cit. '

28) S. Helgason, "Differential geometry and symmetric spaces",

Academic Press , New York (1962), p. 135;

N. Jacobson, loc. cit. , p. 36.

-35-

29) To avoid misunderstanding it is important to notice that the

parity transformation must satisfy (II. 3) for our interpretation

of O(2. 4). It is not an inner automorphism and cannot be

represented by y in the representations A and A since

this would violate the C. R. (II. 3).

30) Y. Murai, Nucl. Phys. 6, 489 (1958).

5)31) R. L. Ingraham, loc. cit.

32) Hepner's treatment has a number of errors which have been

corrected, e. g., the interaction he designates as conformal

invariant (T// y^Vj) * Op y^{*)r^ ) is in fact not so. Hepner

also does not consider the significant rSle of n in (IV. 4) and

(V.2).

33) M. Gell-Mann, Phys. Rev. 1JJ5, 1067 (1962).

34) Note that the o"-"meson" plays two different r61es here.

Firstly it is a manifestation of the breaking of dilatation sym-

metry, eq. (A. 2a). Secondly, it provides an attractive irN

force which is necessary to cancel some of the big s-wave

repulsion inherent in the non-derivative ir-N coupling jwhich

is not observed experimentally. Recall that requirement of

validity of eq. (A. 2b) does not allow for a derivative irN

coupling. (Sec. III..)

-36-

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